The Italians went into the clash at Wembley needing victory or at least a 3-3 draw to advance to the last eight.
And their task was complicated six minutes before the pause when Son Heung-Min scored his 16th goal of the season to give the hosts the lead.
With Spurs seemingly on course for their first appearance in the last eight since 2011, Gonzalo Higuain punished poor marking to steer in Juventus's equaliser after 64 minutes and Paulo Dybala struck three minutes later.
"We are used to playing these games," said Allegri. "We can go in with a clear head and be calm. We are improving as a team in Europe and a performance like this proves that."
Allegri added: "Who knows, anything can happen in the Champions League. We are in the last eight. You look around and there are teams like Real Madrid who could do some damage. That's not to say we can't compete with these teams."
Mauricio Pochettino, Allegri's Tottenham counterpart, rued his side's failure to capitalise on their chances.
"If Harry Kane scored at the end or we scored twice in the first half, maybe we are talking differently," he said.
"I'm happy with my players. We did everything to try to win. It was not a lack of experience, not a lack of concentration. We conceded three chances and they scored twice. We had a lot of chances and only scored one.
"We can talk about a lot of situations but sometimes you need some luck to win."